Collective Edge Modes near the onset of a graphene quantum spin Hall state

Graphene subject to a strong, tilted magnetic field exhibits an insulator-metal transition tuneable by tilt-angle, which is attributed to the transition from a canted antiferromagnetic (CAF) to a ferromagnetic (FM) bulk state at filling factor $\nu=0$. We develop a theoretical description for the spin and valley edge textures in the two phases, and the implied evolution in the nature of edge modes through the transition. Based upon numerical Hartree-Fock calculations, we derive a simple description of the spin-valley domain wall for arbitrary Zeeman energy $E_z$, parameterized by $two$ canting angles. Low-energy charged excitations can be constructed by imposing a slowly varying spin rotation on this state. In the CAF, these involve binding a vortex (meron) of the bulk state to a spin twist at the edge, so that the $bulk$ spin stiffness controls the excitation energy. As the CAF-FM transition is approached ($E_z\rightarrow E_z^c$), the bulk stiffness vanishes linearly with $(E_z^c-E_z)$ and the vortex unbinds from the edge, yielding a gapless edge excitation characteristic of a quantum spin Hall state. Our model predicts the $E_z$-dependence of the activation gap in edge transport, and offers a qualitative picture of how this transport should evolve with filling factor.